Relative CW approximation, Hurewicz fibrations and Serre fibrations Let $B$ be a finite CW complex and $p:E\to B$ a Hurewicz fibration. 
I know that every topological space $E$ has the weak(!) homotopy type of a CW complex $D$ realized by a weak equivalence $f:D\to E$. (I apologize for asking two things in one question but they are so related that splitting them up seems a little exaggerated.)


*

*Is it possible to find a CW complex $D'$, a Hurewicz fibration $p':D'\to B$ and a weak equivalence $f':D'\to E$ such that $p'=p\circ f'$?

*Is it possible to find a Serre fibration $q:C\to B$ and a weak equivalence $g:C\to E$ such that $q=p\circ g$? Can I choose $C$ to be a CW complex?
ADDENDUM:
Due to Henry T. Horton's great answer, the whole question reduces to the following question.
Let $E$ be a topological space. Is it possible to find a CW complex $D$ and a weak equivalence $f:D\to E$ such that $f$ is also a Serre fibration?
 A: For the second question, consider the following. Let
$$B^I \longrightarrow B,$$
$$\alpha \mapsto \alpha(0)$$
be the path space fibration and consider the pullback
\begin{matrix}
p^\ast B^I & \longrightarrow & E \\
 \downarrow & & \downarrow \\
 B^I & \longrightarrow & B
\end{matrix}
Write $P_p = p^\ast B^I$; we call $P_p$ the mapping path space of $p: E \longrightarrow B$. We have the mapping path fibration
$$q: P_p \longrightarrow B,$$
$$(e, \alpha) \mapsto \alpha(1).$$

Theorem. $q: P_p \longrightarrow B$ is a Serre fibration and there exists a homotopy equivalence $g: P_p \longrightarrow E$ such that $q = p \circ g$. In fact, since $p$ is a fibration, $g$ is a fiber homotopy equivalence.

The map $g: P_p \longrightarrow E$ above is actually just the projection onto the first factor:
$$P_p \longrightarrow E,$$
$$(e, \alpha) \mapsto e.$$
See the proof of Theorem 6.18 of Davis and Kirk's Lecture Notes in Algebraic Topology (available here) for the details on why $q$ is a Serre fibration.
